Theorists Without a Theory

In his essay, George Ellis does an excellent job of explaining how some highly publicized speculative claims about theories involving a multiverse have “slipped the leash” of experiment, leading this area of theoretical physics to a strange place. One where the question of what is and what is not science has become open to debate. Here I would like to argue that it is important to recognize the extent to which it is not new and subtle issues about the relation of theory and experiment that are relevant. What is going on is something much simpler: the theorists do not actually have a theory.

In some of the cases mentioned by Ellis this is rather obvious. Examining Max Tegmark’s multiverse of “all mathematical structures,” little mathematical training is needed to see that this is empty as a theory of the physical world.1 There is a long tradition of rumination about possible other universes, but until recently such an activity was typically ignored by physicists as unscientific. This changed back in 2003, with the advent of the “anthropic landscape of string theory,” which claimed that string theory provided a scientific theory of a multiverse.2

To understand what the problems are with this claim one needs to understand what string theory really is, as well as what string vacua are supposed to be. These are extremely complex topics, far beyond what can be addressed here.3 A crude summary of the situation is that a well-defined theory exists for supersymmetric strings propagating in a fixed flat ten space-time dimensional background. This is supposed to be just one limit of a conjectural theory (M-theory) with other conjectured solutions exhibiting four large space-time dimensions. It is these conjectural solutions that are the string vacua, and our physical laws are supposedly determined by the choice of such a solution. All evidence from work on such conjectures is that the known constraints on such string vacua provide no significant predictions about observable physics.

An actual theory of string vacua would characterize them as solutions of some equations defined on some space parametrizing string-theory backgrounds. No such theory exists. Both the space of backgrounds and the equations remain unknown. To get anything that looks like known physics, quite complex choices of background data need to be made, with no indication that one is doing anything other than producing ad hoc ugly constructions designed to match observations, but with no predictive power. While the lack of predictions from this activity is sometimes attributed to the difficulty of the calculations, the real problem is the lack of a theoretical framework capable of giving non-empty results.

Claims are often made that the theory of inflation provides evidence for a multiverse with different physics in each universe. If one looks into actual models of inflation one finds that again, no theory of the sort has been claimed. Most inflationary models involve just a single inflaton field, coupled to the space-time geometry, with nothing in the model able to determine anything about the nature of elementary particles and their behavior. Claims about inflation and a multiverse of different physical laws are based on the hope that the single inflaton field is just one degree of freedom of the space of string backgrounds. There is no viable theory of what this space might be.

The inability of the multiverse paradigm to make any predictions is sometimes attributed to the measure problem: one cannot put a measure on an infinite set giving equal weight to each element. One problem here is the equal weight assumption, which is a reflection of the lack of an actual theory. A well-defined theory would, in principle, allow one to calculate what probabilistic weight to assign to each possibility in the infinite set, giving a consistent measure. Even before getting to this measure problem though, there is a much more serious problem: one does not even know what space it is that one is supposed to be looking for a measure on. One lacks a viable theory that would describe the set of possible universes—the string vacua in the string theory framework—and is thus unable to even specify the measure problem at hand, much less hope to resolve it.

The strongest evidence for a multiverse is generally taken to be the apparently successful Weinberg argument for the anthropic explanation of the value of the cosmological constant (CC).4 Without going into the subtleties of this story, one can see here the lack of any actual physical theory. With no underlying theory in which the statistical distribution of values of the CC is calculable, an assumption is made of a flat distribution; prior to invoking anthropics, any value of the CC is equally likely. As a theory of the physics of the CC, this is effectively exactly the same as my own personal theory, which is that I have absolutely no idea whatsoever about what the physics is determining the CC, so any value is equally likely. An accurate characterization of the situation is that neither I nor those invoking the Weinberg argument have a viable theory of the physics at issue here; there’s no scientific theory to test, so no issue about scientific testability.

Peter Woit

George Ellisreplies:

Peter Woit raises the relation of all this to string theory, which is taken for granted by some multiverse enthusiasts but not others, and demonstrates that there are major issues arising in this context. It is an important addition to what I have written.

Peter Woit is a Senior Lecturer in the Mathematics department at Columbia University.

George Ellis is Emeritus Distinguished Professor of Complex Systems in the Department of Mathematics and Applied Mathematics at the University of Cape Town in South Africa.